The U.S. is considering classifying the country’s election system as part of its “critical infrastructure,” putting it on par with the electrical and banking systems, Obama administration officials said Wednesday.

The comments indicate the federal government is looking at taking a stronger hand in safeguarding the elections process, at a time when cybersecurity attacks have targeted the Democratic Party. Such a move could be a massive undertaking; Department of Homeland Security Secretary Jeh Johnson said elections are currently...